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Question
If A and B are (−2, −2) and (2, −4), respectively, find the coordinates of P such that `"AP" = 3/7 "AB"` and P lies on the line segment AB.
Solution 1
The coordinates of point A and B are (−2, −2) and (2, −4) respectively.
Since AP = `3/7 "AB"`
Therefore, AP: PB = 3:4
Point P divides the line segment AB in the ratio 3:4.
Coordinates of P = `((3xx2+4xx(-2))/(3+4), (3xx(-4)+4xx(-2))/(3+4))`
= `((6-8)/7, (-12-8)/7)`
= `(-2/7, -20/7)`
Solution 2
We have two points A (-2, -2) and B (2, -4). Let P be any point which divides AB as
`"AP" = 3/7 "AB"`
Since,
AB = (AP + BP)
So,
7AP = 3AB
7AP = 3(AP + BP)
4AP = 3BP
`("AP")/("BP") = 3/4`
Now according to the section formula if any point P divides a line segment joining A(x1, y1) and B(x2, y2) in the ratio m: n internally than,
P(x, y) = `((nx_1 + mx_2)/(m + n)"," (ny_1 + my_2)/(m + n))`
Therefore, P divides AB in the ratio 3: 4. So,
P(x, y) = `((3(2) + 4(-2))/(3 + 4)"," (3(-4) + 4(-2))/(3 + 4))`
= `(-2/7,-20/7)`
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Activity:
∴ By section formula,
∴ x = `("m"x_2 + "n"x_1)/square`,
∴ x = `(3 xx 8 + 1 xx 4)/(3 + 1)`,
= `(square + 4)/4`,
∴ x = `square`,
∴ y = `square/("m" + "n")`
∴ y = `(3 xx 5 + 1 xx (-3))/(3 + 1)`
= `(square - 3)/4`
∴ y = `square`
